Quote by Ben Niehoff
This has nothing to do with it. Field lines work because the sourcefree Maxwell equations imply that the 2form F is harmonic.
Harmonic forms (in any dimension) have the distinction that they capture purely topological information. If you integrate a harmonic nform over a closed nsurface, the result is either zero or nonzero, depending on whether the nsurface encloses some topological feature (for example, a 1surface on a cylinder might wrap around the cylinder...or a 2surface in R^3 might enclose a charge). Any nsurface that encloses the same set of topological features must give you the same result.
You can think of this as a higherdimensional analogue of contour integration. In fact, all analytic functions on the complex plane satisfy Laplace's equation, which is why contour integration works.

Just to clarify, by harmonic you mean: [tex](d\delta+\delta d)F=0[/tex] where d is the exterior derivative and delta is the codfifferential?
Is this true? It seems since dF=0 by definition, then we need to show: [tex]d\delta F=d(*d*F)=4\pi(d**J)=0[/tex]
Is it true that [tex]dJ=0[/tex]? I can't think of a reason for this...
EDIT: Wait, is that just conservation of charge? It certainly isn't the usual way of expressing it...and I can't seem think think clearly enough at this late hour to figure this out...lol...